4 Interpolation

For the strict interpolation approach, we note firstly that, if a good approximate form $\Omega^{\rm (approx)}$, representing the collision strength $\Omega$, is available, then a graph of the ratio $[\Omega(X_i)/\Omega^{\rm (approx)}_i$: i=1,...,N] draws attention clearly to errors or misprints in the tabular source data. Also, the interpolating spline through the ratio values (which will then not differ substantially from unity) is accurate. We seek to introduce good approximations to collision strengths based on the following parametrised approximate forms. The approximate forms for neutral species have two parts (- and +) which apply to the threshold and asymptotic regions respectively and the two parts are matched at some tabular value of the collision parameter X=X_k:

4.1 Type 1a

$$\Omega^{\rm (approx)}_+ F_3S\ln(X-X_k+F_2)$$ = (6)

$$\Omega^{\rm (approx)}_- F_1((X-1)/(X_k-1))^B\exp(-(B-b')$$ = ((X-X_k)/(X_k-1)))

where b'=(X_k-1)F₃S/F₁F₂ and $F_1=\Omega_k$.

4.2 Type 2a

$$\Omega^{\rm (approx)}_+$$ = F₂+F₃/(X-X_k+1) (7)

$$\Omega^{\rm (approx)}_- F_1((X-1)/(X_k-1))^B\exp(-(B-b')$$ = ((X-X_k)/(X_k-1)))

where b'=(X_k-1)F₃/F₁ and $F_1=\Omega_k$.

4.3 Type 3a

$$\Omega^{\rm (approx)}_+$$ = F₃/(X-X_k+F₂)² (8)

$$\Omega^{\rm (approx)}_- F_1((X-1)/(X_k-1))^B\exp(-(B-b')$$ = ((X-X_k)/(X_k-1)))

where b'=-2(X_k-1)F₃/F₁F₂³ and $F_1=\Omega_k$.

4.4 Type 4a

$$\Omega^{\rm (approx)}_+$$ (9)

$$\Omega^{\rm (approx)}_-$$

We note the inclusion of the line strength in the definition of type 1. If high energy data is correct and consistent with the line strength then F₃should be equal to 4/3. Burgess & Tully impose this condition on their optimised fit. Type 4 is the case of no approximate form. There is no adjustable transformation of the independent variable X then but, as in the Burgess-Tully case, it is convenient to transform to X'=1-2/X for displaying the ratio over the finite range [-1, 1] with a balanced weighting between the threshold and asymptotic regions. The parameters of the approximate forms are determined by fixing $\Omega^{\rm (approx)}_-(X_0) =\Omega_0$, $\Omega^{\rm (approx)}_+(X_N) =\Omega_N$ and $\Omega^{\rm (approx)}_+(X_k)=\Omega^{\rm (approx)}_-(X_k)=\Omega_k$. Note that the $\Omega$ values at X_N and X_k fix F₂ and F₃, explicitly for types 2a and 3a and implicitly for type 1a. Then b' is determined explicitly from continuity of the derivative at X_k, that is ${\Omega^{\rm (approx)}_+}'(X_k)={\Omega^{\rm (approx)}_-}'(X_k)$, so that an implicit equation is left finally for determining B. The procedure requires $N \ge 3$. For N=2, we force F₃=4/3 for type 1, F₃=0 for type 2 and F₂=1 for type 3. For N=1, we force B=0.5. Also, data which do not manifest the expected monotonic asymptotic behaviours can prevent solution for the approximate form. The forced contractions given above are again imposed in such cases. For historical reasons, in the ADAS graphical user interface, the normal solution for the approximate form is called "2-point'' fitting and the contracted problem cases, "1-point'' fitting. For $N \ge 4$, there is a choice of matching position. We seek X_k with $2 \leq k \leq N-1$ so that $\sum_{i=1}^N(\Omega(X_i)-\Omega^{\rm (approx)}(X_i))^2$is a minimum. If k=1, the tabular data set does not span the threshold region effectively and so we set B=0.5. If k=N, the tabular data set does not span the high energy region effectively and so we set F₃=(4/3). Figures 2b and 3b show the interpolation of the $\Omega /\Omega ^{\rm (approx)}$ ratio graph. The data are for the same examples as in Figs. 2a and 3a respectively. The cases are again analysed as types 1a and 3a.

Figure 3: a) Type 3a: B-C nine-knot spline least squares fit to the $\rm HeI(1~^1S - 2~^3P)$spin-change electron-impact collision strength data of de Heer et al. (1992). B=0.685, C=1.191; b) Type 3a: $\Omega /\Omega ^{\rm (approx)}$ ratio plot and interpolating spline for the same spin-change collision strength data. B=0.56483, $b^{\prime }=-1.21690$, F1=0.04090, F2=3.84420, F3=0.60442, Xk=3.33900. The stated uncertainty of the data is $\sim$10%

Figure 4: ${\rm H}_2(X~^1\Sigma^+_g)+{\rm e} \rightarrow {\rm H}_2(d~^3\Pi^+_u)+{\rm e}$. a) Type 3a interpolative ratio plot, B=2.65490, $b^{\prime }=-1.01780$, F1=0.04910, F2=1.30870, F3=0.08409, Xk=1.66600; b) B-C plot with original data, B=0.778, C=2.982; c) B-C plot after adjustment of first and last two data values, B=0.778, C=2.940

Figures 4a-c show the $\Omega /\Omega ^{\rm (approx)}$ ratio and B-C plots for the spin change hydrogen molecular reaction cross-section $H_2(X~^1\Sigma^+_g)+{\rm e} \rightarrow H_2(d~^3\Pi^+_u)+{\rm e}$. The source data are from Behringer & Fantz (1996) and are themselves a fit to very limited experimental data (Mohlmann & De Heer 1976). The ratio plot seems to indicate that the first data value is too low, while the last two data values suggest that there is some deviation (again too low) from the expected high energy behaviour. The B-C plot for these data (Fig. 4b) shows that the spline diverges at both limits for reasonable values of the parameters. In Fig. 4c, the first and last two data points have been adjusted (see Sect. 6 below) and an optimised fit then obtained. These adjustments are well outside the range of the original experimental data and perhaps deliver a safer extrapolation in the limiting asymptotic and threshold regions.

Figure 5: An editable graph window with an active B-C plot. The number pair at the bottom of the displayed graph shows the cursor position, in graph coordinates, when a point is being moved