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3 The effect of resonances

Resonances appear as spikes or dips on the graph of $\sl \Omega$. The spikes are sometimes isolated but more often they come as a dense forest of peaks. Resonances can have a big effect on $\sl \Upsilon$, especially when the transition is optically forbidden, and may cause $\sl \Upsilon$ to be between a few per cent or several factors larger than the predictions of distorted wave approximations. For the purpose of comparison we list in Table 5 results for ${\sl \Upsilon}(1-2)$ based on collision strengths from (a) the IRON Project, (b) the IRON Project (chopped) and (c) a distorted wave approximation. The collision strength in (b) was obtained by imposing an arbitrary maximum peak height of $\rm 1.25\,10^{-3}$ which effectively chops the tops off the resonance peaks. The distorted wave collision strength used in (c) is from Bhatia & Mason (1986).
  
Table 5: Showing the effect on ${\sl \Upsilon}(1-2)$ of chopping off the IRON Project resonances: (a), IRON; (b) IRON (chopped); (c) Bhatia & Mason (1986)

\begin{tabular}
{llll}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
log$T$\s...
 ...9 & 0.26 & 0.24 \\ \noalign{\smallskip}
\hline
\noalign{\smallskip}\end{tabular}


  
Table 6: Showing how the high energy contribution to $\sl \Upsilon$ increases with temperature for three types of transition. Intersytem (non electric dipole) transition: (a) ${\sl \Upsilon}(1-2)$ with $E_{\rm max}=346.8\,{\rm Ry}$; (b) ${\sl \Upsilon}(1-2)$ with $E_{\rm max} = 10^5\,{\rm Ry}$. Intersystem (electric dipole) transition: (c) ${\sl \Upsilon}(1-3)$ with $E_{\rm max}=346.6\,{\rm Ry}$; (d) ${\sl \Upsilon}(1-3)$ with $E_{\rm max} = 10^5\,{\rm Ry}$. Electric dipole transition: (e) ${\sl \Upsilon}(2-7)$ with $E_{\rm max}=340.7\,{\rm Ry}$; (f) ${\sl \Upsilon}(2-7)$ with $E_{\rm max} = 10^5\,{\rm Ry}$. $E_{\rm max}$ is the value used for the upper limit in the integral that defines $\sl \Upsilon$ and it should in theory be $\infty$. $\,(2.01^{-3} \equiv \, 2.01 \ 10^{-3})$

\begin{tabular}
{lllllllllll}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
l...
 ...$^{-1}$\space \\  
\noalign{\smallskip}
\hline
\noalign{\smallskip}\end{tabular}

As mentioned above, we use the Breit-Pauli R-matrix code for collision energies up to 103.05816 Ry after which we replace it by the simpler LS coupling code together with the algebraic code JAJOM (Saraph 1978). In this way we are able to extend the Breit-Pauli results to higher energies by running the LS code at 116, 127.5, 170, 250, 350 Ry. This sparse mesh is ample for our purposes since no resonances are encountered over this energy range. It is difficult to go much beyond 350 Ry using the R-matrix codes. In order to obtain meaningful and reliable thermally averaged collision strengths at the high temperatures given in Table 9 it is necessary to know $\sl \Omega$ at energies of one or two thousand rydbergs, i.e. well beyond 350 Ry. We are able to make reasonable extrapolations of our data by means of the computer program OmeUps (Burgess & Tully 1992).


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