2 The transformations

We note that the coordinate transformations used in the Burgess & Tully method have a close connection with the general mathematical shape of the underlying collision strengths, so some commonality of the optimised fitting and strict interpolation approaches can be promoted by making the coordinate transformations and the approximate forms analytically similar. We shall continue to use the Burgess & Tully transformations in this work and their distinction of collision strength types by a number (1 $\equiv$ dipole; 2 $\equiv$ non-dipole, non-spin change; 3 $\equiv$ spin change; 4 $\equiv$ weak dipole) but they will apply strictly to the high energy region. We introduce new transformations for the threshold region and these will be distinguished by a letter. The transformations must satisfy certain monotonicity and continuity constraints - the latter at a switching energy where the low energy and high energy transformations meet. Then we require related approximate forms for the low and high energy regions which can again match smoothly at a switching energy.

The shapes of the energy transformations and approximate forms for the mapping and representation of collision strengths in the low energy (threshold) and high energy (asymptotic) regions which have proved useful and relevant to our further discussion are illustrated in Fig. 1. The dashed lines relate to transformations and the solid lines to approximate forms. The dashed curve (a'), with $B~\sim~0.5$ (B is the threshold power law - cf. Eq. (1)) is the simplest monotonic form for the threshold behaviour of s-wave dominated collision strengths. Its curvature allows gradient matching with the energy transformation of logarithmic character for dipole allowed collision strengths of the Burgess-Tully type 1 and it is represented here by the solid line asymptotic form labelled 1. For $B~\ge~1.0$, the positive curvature is unsuitable for matching. The dashed curve (a) is also monotonic with the same threshold power law and is compatible with the energy transformations used in all of the Burgess-Tully types. The characteristic of non-dipole and spin-change collision strengths in the asymptotic region (solid lines (2)and (3) respectively) is to show negative gradients and the solid line threshold form (a) with B = 1.5 is provided for matching in this case. The energy threshold transformation, dashed line (b), is an alternative to (a) but with an exponential character. We prescribe these transformations in detail below and the approximate forms in Sect. 4.

Figure 1: The interval [-1, 0] shows threshold behaviours suitable for energy transformations (dashed) and approximate forms (solid), labelled by letter and parameter B (see Sect. 2). The interval [0, 1] shows similarly the asymptotic transformations and forms, labelled by numerals. The curves are unmatched between the threshold and asymptotic regions only because they have been normalised to unit peak value and/or adjusted to emphasis the shapes. X is the threshold scaled energy parameter. The labelling follows the classification scheme given in Sect. 2

Let the transition energy between lower level i and upper level f be E_if. The threshold scaled energy parameter is defined by X=(E_f/E_if)+1. Working in energies E_f relative to the threshold energy, let E₀ mark the division between the threshold region and the high energy region. We call this the switching energy. We seek mappings x_- and x₊ of the independent variable from final energies [0,E₀] to [-1, 0] and from final energies $[E_0,\infty]$ to [0,1] respectively and compatible transformations of the collision strengths. We introduce a scale parameter, B, for the threshold region and C, for the high energy region. C is the Burgess-Tully parameter. The transformations are given below, indexed by Burgess-Tully type (see Burgess & Tully 1992) and a letter for the threshold type.

2.1 Type 1a

$$1-\ln(C)/\ln((E_f-E_0)/E_{if}+C)$$ x₊ =

x_- = 2(E_f/E₀)^B/((E_f/E₀)^B+1)-1

$$\Omega/\ln((E_f-E_0)/E_{if}+{\rm e})$$ y₊ = (1)

$$\Omega/((E_f/E_0)^B\exp(-(B-E_0/{\rm e}E_{if})$$ y_- = ((E_f/E₀)-1)))

$$\Omega(E_0)$$ y₊(0) =

$$4\omega_if_{if}/E_{if} .$$ y₊(1) =

2.2 Type 2a

x₊ = ((E_f-E₀)/E_if)/(((E_f-E₀)/E_if)+C)

x_- = 2(E_f/E₀)^B/((E_f/E₀)^B+1)-1

$$\Omega$$ y₊ = (2)

$$\Omega/((E_f/E_0)^B\exp(-B((E_f/E_0)-1)))$$ y_- =

$$\Omega(E_0).$$ y₊(0) =

2.3 Type 3a

x₊ = ((E_f-E₀)/E_if)/(((E_f-E₀)/E_if)+C)

x_- = 2(E_f/E₀)^B/((E_f/E₀)^B+1)-1

$$((E_f-E_0)/E_{if})+1)^2\Omega$$ y₊ = (3)

$$\Omega/((E_f/E_0)^B\exp(-(B+2E_0/E_{if})$$ y_- = ((E_f/E₀)-1)))

$$\Omega(E_0).$$ y₊(0) =

2.4 Type 4a

$$1-\ln(C)/\ln((E_f-E_0)/E_{if}+C)$$ x₊ =

x_- = 2(E_f/E₀)^B/((E_f/E₀)^B+1)-1

$$\Omega/\ln((E_f-E_0)/E_{if}+C)$$ y₊ = (4)

$$\Omega/(\ln(C)(E_f/E_0)^B\exp($$ y_- =

$$-(B-E_0/C\ln(C)E_{if})((E_f/E_0)-1)))$$

$$\Omega(E_0)/\ln(C)$$ y₊(0) =

$$4\omega_if_{if}/E_{if} .$$ y₊(1) =

An alternative transformation of the independent variable in the threshold region is also provided which is of exponential rather than power law type.

2.5 Type b

$$\exp(-B[\sqrt{(E_{if}/E_f)}-\sqrt{(E_{if}/E_0)}])-1.$$ x_- = (5)

These transformations introduce three parameters, B, C, and E₀. We impose a requirement of continuity on the derivative of x(E_f) at x=0 which determines E₀ in terms of B and C. For example, for types 1a and 4a, $E_0=E_{if}BC\ln(C)/2$; for types 2a and 3a, E₀=E_ifBC/2; for types 1b and 4b, $E_0=E_{if}(BC\ln(C)/2)^{2/3}$; for types 2b and 3b, E₀=E_if(BC/2)^2/3. Thus the Burgess-Tully C parameter allows the usual distortion of the high energy x scale while the B parameter allows stretching or contraction of the threshold region. Note that the transformations maintain continuity of value and derivative of x_- and x₊ at E_f=E₀ and also of y_- and y₊. The stretching enabled by the variation of B and C allows convergence on the best matching and fit of the threshold and asymptotic regions.