3 The BT method

In order to analyse collisional data and to present them in compact form, BT introduce a scaling procedure where the collision strength $\Omega(E)$ is mapped onto the reduced form $\Omega_{\rm r}(E_{\rm r})$, where the infinite energy E range is scaled to the finite $E_{\rm r}$ interval (0,1). For a quadrupole transition, such as $\sp1$D $-\sp1$S in Ar III, the BT scaling prescription is given by

$$\begin{eqnarray} & E_{\rm r} &=\frac{E}{\Delta E}\left(\frac{E}{\Delta E}+C\right)\sp{-1}\\ & \Omega_{\rm r}(E_{\rm r}) &=\Omega(E)\end{eqnarray}$$ (1) (2)

with $\Delta E$ being the transition energy, E the electron energy with respect to the reaction threshold and C an adjustable scaling parameter. A key aspect of the BT approach lies in the fact that the limiting points $\Omega_{\rm r}(0)$ and $\Omega_{\rm r}(1)$ are both finite and can be computed. BT have discussed that for a quadrupole transition these points are

$$\begin{eqnarray} &\Omega_{\rm r}(0)&=\Omega(0) \\ &\Omega_{\rm r}(1)&=\Omega_{\rm CB}\end{eqnarray}$$ (3) (4)

where $\Omega_{\rm CB}$ is the Coulomb-Born high-energy limit. This formalism can also be extended to treat the effective collision strength

$$\Upsilon(T)=\int_0\sp\infty\Omega(E) \exp(-E/\kappa T){\rm d}(E/\kappa T)\ ,$$ (5)

through the analogous relations

$$\begin{eqnarray} &T_{\rm r} &=\frac{\kappa T}{\Delta E}\left(\frac{\kappa T}{\Delta E}+C\right)\sp{-1} \\ &\Upsilon_{\rm r}(T_{\rm r})&=\Upsilon(T)\end{eqnarray}$$ (6) (7)

where T is the electron temperature and $\kappa$ the Boltzmann constant; the limiting points now become

$$\begin{eqnarray} &\Upsilon_{\rm r}(0)&=\Omega(0) \\ &\Upsilon_{\rm r}(1)&=\Omega_{\rm CB} \ .\end{eqnarray}$$ (8) (9)

A second important point in the BT approach is that the reduced effective collision strength can be neatly fitted in its entire range. A 5-point spline is usually sufficient, and thus leads to a notably compact way of presenting collisional data.

With specific reference to the questioned transition in Ar III, BCT have computed a Coulomb-Born high-energy limit of $\sim\!1.64$. This computation includes configuration interaction effects that are found only to cause changes of a few per cent. Then, in a plot of the IP-X effective collision strengths, they scale the fairly low temperature range of 0 - 10⁵ K to occupy 90% of the reduced temperature interval and they find an extrapolated high-energy limit of $\sim\!0.8$, a factor of 2 lower than the Coulomb-Born limit, which led them to question the accuracy of the calculation.