Generalized oscillator strengths for the(1s22s2)S1→(1s22s2p)P1transition in some Be-like ions

Generalized oscillator strengths $f\left(K\right)\mathrm{}$ have been calculated for the $\left(1\mathrm{}{s}^{2}2s^2\right){}_{}{}^1{}_{}{}^{}S\to \left(1\mathrm{}{s}^{2}2s2p\right){}_{}{}^1{}_{}{}^{}P$ transition of the Be-like ions when $Z=4-8\mathrm{}$. A "length" formula and two versions of a "velocity" formula for $f\left(K\right)\mathrm{}$ were used in each instance. For each ion, $K$- and $L$-intrashell pair correlation functions were included in the description of the ground-state wave function, and the $K$-shell pair function was also introduced into the excited state. The pair functions were determined after the general manner proposed by Sinanoğlu. For $K=0\mathrm{}$, $f\left(K\right)\mathrm{}$ is equivalent to the optical oscillator strength and our results can therefore be compared with several other theoretical values and with experiment. The excellent agreement of our $f\left(0\right)\mathrm{}$ values when compared with experiment, coupled with the high degree of consistency amongst the $f\left(K\right)\mathrm{}$ values when evaluated, in turn, from the three formulations mentioned above, suggests that our correlated results should possess considerable reliability, particularly at the lower end of the $Z$ range. The sensitivity of the $f\left(K\right)\mathrm{}$ values to the presence of electron correlation effects has been examined by repeating the calculations using Hartree-Fock descriptions for both the ground and excited states. Comparison between our correlated and noncorrelated results revealed that the presence of the pair functions was indispensable in order to achieve agreement between $f\left(K\right)\mathrm{}$ values derived from the "length" and "velocity" formulations. For Be, the $f\left(K\right)\mathrm{}$ values were used to obtain total inelastic scattering cross sections, within the first Born approximation, for both proton and electron impact. Although our allowance for electron correlation produced significant changes in the "length"-based cross sections, "velocity" values proved to be considerably less sensitive over a large energy range for both types of projectile.