2 Compacting atomic collision data

We use notation as established in B&T. All the transitions treated in the present paper are of type 2. In this section we specify the changes required for these transitions as a consequence of including relativistic effects. We use the same equation numbers as in B&T, with * added to the number of the modified equation. Derivations, and details for other types of transition, will be given elsewhere.

In order to distinguish the adjustable parameter C occurring in the different fits we denote it by C_E, C_T, C_Z according to whether the fit is over the range of energy E, temperature T or ion charge Z.

2.1 Fitting the collision strengths ${\sl \Omega}$

Equation (9) of B&T is unchanged except that C is denoted by C_E, but (10) becomes

$y(x)= {\sl \Omega} / ( 1 + \epsilon \, E_i)( 1 + \epsilon \, E_j) \, ,$(10^*)

where $\epsilon = I_{\infty} / m c^2 = \alpha^2/2 = 2.6626 \ 10^{-5}$, $\alpha$ being the fine structure constant.

2.2 Limiting values of ${\sl \Omega}(E_j)$ when $E_j \to \infty$

We use the interactive FORTRAN program BORN written by Burgess (1998) to compute the non relativistic high energy limit points presented in Table 1.

  

Table 1: High energy Born collision strengths ${\sl \Omega}(nlj)$ for transitions ${\rm 2s_{1/2}} \to nlj$

We take radial orbitals $P_{\rm 1s}\,, P_{\rm 2s}\,$ from Clementi & Roetti (1974) and calculate $P_{\rm 3s}\,, P_{\rm 3d}\,$, $P_{\rm 4s}\,, P_{\rm 4d}\,$, $P_{\rm 5s}\,, P_{\rm 5d}\,$ using Hibbert's (1975) program CIV3. The parameterised form of P_nl(r) is

$$P_{nl}(r) = \sum_{p=1}^kc_{nlp} \Bigl[{{(2\zeta_{nlp})^{2q_{... ..._{nlp})!}}\Bigr]^{1/2}r^{q_{nlp}}\exp(-\zeta_{nlp} r). \eqno(1)$$

Our choice for the integers $(k,\, q_{nlp})$ in Eq. (1) is $(n,\,p)$ for l = 0 and $(n-2,\, p+2)$ for l = 2. Results for four ions in the sequence, namely Z₀ = 8, 16, 26, 36, are given in Table 2.

  

Table 2: Radial orbital parameters $\zeta_{nlp}(Z_0)$ and c_nlp(Z₀)

We obtained the values for $\zeta_{nlp}$ and c_nlp by using CIV3 to minimise the ${\rm 1s}^{2}nl$ term energy. Since in general many local minima exist it can be quite an arduous task locating the one that seems to be the lowest. A similar investigation has in the past been carried out for some of the orbitals included here (see Tully et al. 1990 and Berrington & Tully 1997). A comparison of the present and past data shows that they are not always the same. Although we have not compared the orbitals it is our belief that they do not differ significantly. The numerical differences alluded to here reflect the fact that in the present work we have given greater attention to locating the term minima than in the past. In all cases we use the minimization code MODDAV by setting the CIV3 parameter IDAVID = 0. We assume each orbital P_nl to have the minimum number of exponents dictated by nl and therefore the coefficients c_nlp are determined by the conditions of orthonormality, but for convenience we include their numerical values in Table 2.

2.3 Values of ${\sl \Omega}$ when Z is large

It is extremely useful, when fitting collision data along an entire isoelectronic sequence, to have an indication of the behaviour of the collision strengths when Z becomes large. The true limiting values do not exist since the relativistic wave functions for the ion collapse when Z is too large (e.g. for s-states, when $Z \to 137$). Nevertheless, the non-relativistic values, scaled by a factor (Z+1)², do exist in the limit and are useful in this context.

D. H. Sampson and his co-workers have, over the past two decades, produced an enormous amount of reliable collision data in just this limit, and we make use of their Coulomb Born Oppenheimer results for the hydrogenic transitions $\rm 2s \to 3s,4s,5s$ and $\rm 2s \to 3d,4d,5d$. These are available in Golden et al. (1981) and Clark et al. (1982). However it should be noted that the values they tabulate for the collision strength need to be multiplied by a factor 2 in order to be consistent with the definitions of ${\sl \Omega}$ and statistical weight given in equations (2) and (3) in Golden et al. (1981).

We have made only approximate allowance for the relativistic corrections to these limiting values. However, the uncertainties arising from this affect only values beyond the range of the ZSF tabulation (i.e. for Z > 89 or E_j > 10⁴ Ryd, T > 10⁹). Further work on this will be given elsewhere.

2.4 Calculation of the effective collision strength ${\sl \Upsilon}(T)$

The definition of ${\sl \Upsilon}$ in terms of ${\sl \Omega}$, Eq. (21) of B&T, is unchanged, (as is the detailed balance Eq. (22)), so that the OmeUpZ program can be used to calculate ${\sl \Upsilon}$ from the fitted ${\sl \Omega}$ as described in Sects. 6.3 and 7.2 of B&T. However, Eq. (20) for the rate coefficient is altered by replacing the factor ${\sl \Upsilon}$, on the right hand side, by ${\sl \Upsilon}/{\sl \Phi}$, where

$${\sl \Phi}=(2mc^2 / \pi kT)^{1/2} \, \exp(mc^2 /kT) \; K_2 (mc^2 /kT) \, , \eqno(\rm 20^*)$$

K₂ being a modified Bessel function (Abramowitz & Stegun 1965). This result is for a low density, high temperature plasma corresponding to the nondegenerate Maxwell-Juttner case discussed by Chandrasekhar (1957, pp. 394-8).

Except for very large values of T, we have ${\sl \Phi} \approx 1 + (15/8)(kT/mc^2) = 1 + 3.1619 \ 10^{-10} \, T$.

2.5 Fitting ${\sl \Upsilon}$ as a function of T

Equation (27) of B&T is unchanged except that C is denoted by C_T, but (28) becomes

$$y(x)= {\sl \Upsilon}/ (1 + \epsilon E_{ij} + \mu T (2 + \epsilon E_{ij} + 2 \mu T ) ) \, , \eqno(\rm 28^*)$$

where $\mu = k / m c^2 = 1.6864 \ 10^{-10}$.

2.6 Fitting the nodal values of ${\sl \Omega}_{\rm r}$ and ${\sl \Upsilon}_{\rm r}$ as functions of $Z_{\rm r}$

As described in Sects. 7.5 and 8.2 of B&T, for a given transition, the standard 5-point cubic spline fitting procedure produces 5 nodal values of ${\sl \Omega}_{\rm r}$ or ${\sl \Upsilon}_{\rm r}$ for each Z, which may themselves then be fitted as a function of a reduced Z variable, so as to produce the doubly reduced quantities ${\sl \Omega}_{\rm rr}$ or ${\sl \Upsilon}_{\rm rr}$. As in B&T, use of a common value of C_E or C_T for all the values of Z is obviously most convenient and leads to no significant loss of accuracy.

  

Table 3: Program data to obtain ${\sl \Omega}$ for all charges $8\leq Z_0 \leq 92$

  

Table 3: continued

Equations (58) and (59) of B&T are unchanged except that C is denoted by C_Z.

This gives a $5\times5$ matrix S_mn of nodal values of ${\sl \Omega}_{\rm rr}(E_{\rm r},Z_{\rm r})$ or ${\sl \Upsilon}_{\rm rr}(T_{\rm r},Z_{\rm r})$ from which ${\sl \Omega}$ or ${\sl \Upsilon}$ may be obtained for all energies or temperatures and all charges.

Values of the S_mn matrix elements, together with the values of C_E or C_T and C_Z are given in Tables 3 and 4.

The fitted values of ${\sl \Omega}$ given by Table 3 were compared with all the original tabulated values given by ZSF for these transitions (i.e. for all 6 energies $\times$ all 85 charges $\times$ 9 transitions); the rms error was 0.3% and the maximum error was 0.9%. The rms error involved in fitting ${\sl \Upsilon}$, to obtain Table 4, was also found to be about 0.3%.

  

Table 4: Program data to obtain ${\sl \Upsilon}$ for all charges $8\leq Z_0 \leq 92$

  

Table 4: continued

As noted in Sect. 2.3, there is some uncertainty in the values of ${\sl \Omega}$ beyond the range of the ZSF tabulation. We estimate that the resulting uncertainty in values obtained from Table 3 and 4 exceeds 1% only if Z₀ > 92 or E_j > 10⁴ Ryd, T > 10⁹).