2 Generalized oscillator strengths

The formulas for computing the generalized oscillator strengths (GOS) are based on the first Born approximation and the LS-Coupling scheme, and may be found in earlier work by the present author ([Ganas 1998]). The following notation appears in the subsequent discussion. The quantity x = K²a₀², where K is the momentum tranfer. Also $x_{\rm t} = W/R$ where W is the excitation energy in eV and R is the Rydberg energy. The quantity $\xi = x/x_{\rm t}$.

Plots of the GOS are shown in Fig. 1. As $\xi$ increases, the GOS for the transitions 3p - ns decrease in oscillatory fashion, while the GOS for 3p - nd decrease monotonically. The GOS for 3p - np also decrease monotonically, but pass through a maximum first.

 

Figure 1: GOS for 3p - ns, 3p - np, 3p - nd transitions in Si I as functions of reduced square of momentum transfer. The curves are the result of the present work. The solid dots, open circles, and triangles are representative fits using Eqs. (2) and (3). Values of the GOS below 10-5 have been multiplied by 105

The calculation of cross sections can be facilitated by parametrizing the GOS with simple analytic forms:

$$f(\xi) = A~({\rm e}^{-\alpha\xi}~+~\beta\xi {\rm e}^{-\gamma\xi})^2 \qquad\mbox{for 3p~$-$~ns, 3p~$-$~nd}$$ (2)

$$f(\xi) = \xi A~({\rm e}^{-\alpha\xi}~+~\beta\xi {\rm e}^{-\gamma\xi})^2 \qquad\mbox{for 3p~$-$~np}.\qquad$$ (3)

The quantities A, $\alpha$, $\beta$, $\gamma$ in Eqs. (2) and (3) are adjustable parameters which are varied so as to obtain the best fit to the numerically generated GOS. The quantity A in Eq. (2) is the oscillator strength, which is the limit of the GOS as $\xi \rightarrow 0$. When using Eq. (2), A is set equal to the value of the oscillator strength, and three-parameter fits are obtained. Four-parameter fits are obtained when using Eq. (3). The values of the four parameters which yield the best fits are given in Table 1. Sample fits are shown in Fig. 1. Equations (2) and (3) actually only reproduce the GOS when it is significantly large.

 

Table 1: Values of the dimensionless parameters A, $\alpha$, $\beta$, $\gamma$ in Eqs. (2) and (3)

 

Figure 2: Integrated cross sections for 3p - ns, 3p - np, and 3p - nd excitations in Si I versus electron impact energy