5 Quadratures

The production of Maxwell-averaged collision strengths is straightforward. The Burgess-Tully Gauss-Laguerre based approach remains valid for the high energy region. The finite range of the low-energy region is treated similarly but with the Gaussian quadrature, in this case tuned to the usual $\sqrt{X-1}$ behaviour. The switch in energy between the two quadratures is not set at the E₀ of Sect. 2 or the X_k of Sect. 4 but, rather, where the threshold collision strength and Maxwellian exponential energy-scale lengths are equal. For the strict interpolative approach, it is noted that the approximate forms are integrable in terms of known functions. Thus, setting $a_{\rm T}=(E_{if}/I_{\rm H})(I_{\rm H}/kT_{\rm e})$ with $I_{\rm H}$ the Rydbery energy unit, the Maxwell averaged collision strength is

$$\Upsilon^{\rm (approx)}(T_{\rm e})=\Upsilon^{\rm (approx)}_-(T_{\rm e})+\Upsilon^{\rm (approx)}_+(T_{\rm e}).$$ (10)

5.1 Type 1a

$$\Upsilon^{\rm (approx)}_-(T_{\rm e}) F_1(X_k-1)a_{\rm T}$$ =

$${\rm e}^{B-b'}\frac{1}{\vert A\vert^{B+1}} \gamma(B+1,A)$$

$$\Upsilon^{\rm (approx)}_+(T_{\rm e}) {\rm e}^{-(X_k-1)a_{\rm T}}F_3S_{if}$$ = (11)

$$[\ln(F_2)+{\rm e}^{a_{\rm T} F_2}E_1(a_{\rm T} F_2)]$$

where $A=(B-b')+(X_k-1) a_{\rm T}$, $\gamma$ is the incomplete gamma function and E₁ is the first exponential. Note that in $\gamma$, the principal value of the logarithm must be taken for negative values of the argument A.

5.2 Type 2a

$$\Upsilon^{\rm (approx)}_-(T_{\rm e}) F_1(X_k-1)a_{\rm T}$$ =

$${\rm e}^{B-b'}\frac{1}{\vert A\vert^{B+1}} \gamma(B+1,A)$$

$$\Upsilon^{\rm (approx)}_+(T_{\rm e}) {\rm e}^{-(X_k-1)a_{\rm T}}$$ = (12)

$$[F_2+F_3{\rm e}^{a_{\rm T}}E_1(a_{\rm T})].$$

5.3 Type 3a

$$\Upsilon^{\rm (approx)}_-(T_{\rm e}) F_1(X_k-1)a_{\rm T}$$ =

$${\rm e}^{B-b'}\frac{1}{\vert A\vert^{B+1}} \gamma(B+1,A)$$

$$\Upsilon^{\rm (approx)}_+(T_{\rm e}) {\rm e}^{-(X_k-1)a_{\rm T}}F_3 a_{\rm T}^2$$ = (13)

$$\frac{1}{(a_{\rm T} F_2)}[1-a_{\rm T} F_2 {\rm e}^{a_{\rm T} F_2}E_1(a_{\rm T} F_2)].$$

Since the quadrature in the threshold region is tuned to the $\sqrt{X-1}$behaviour, modest improvement in the threshold region is achieved by numerical integration of the difference of the interpolated exact data and the approximate form and then completing the integral with the analytic integral of the approximate form. For neutral atoms and molecules in plasmas, distortions of the electron distribution function are common and so dedicated quadrature routines for particular cases are usually required. The optimised spline and fitting algorithms given here are readily integrated into such routines.