Up: : Thermally averaged electron

# 4 Thermal averaging collision strengths

The CCC program furnishes the cross section as a function of Ei, the collision energy relative to term i. We convert this to a table of as a function of Ej which we then use to calculate from Eq. (2) by means of the linear interpolation technique (Burgess & Tully 1992). The upper limit of the integral (2) is replaced by the value of Ej corresponding to the highest value of Ei at which the cross section is calculated by the CCC program. We checked to make sure that the contribution from higher energies was negligible. In order to extend our tabulation of to temperatures well beyond 500000 degrees it will be necessary to calculate or estimate collision strengths at higher energies than those considered here. Alternatively one could make use of the technique developed by Burgess & Tully (1992) in order to extrapolate the present data up to higher temperatures.

 Transition n=2 n=3 n=4 n=5 Transition n=2 n=3 n=4 n=5 3.75 3.075-2 8.431-3 2.397-3 1.000-3 6.198-2 1.642-2 5.317-3 2.593-3 4.00 3.492-2 8.308-3 2.505-3 1.084-3 6.458-2 1.584-2 5.233-3 2.542-3 4.25 3.840-2 8.364-3 2.642-3 1.189-3 6.387-2 1.530-2 5.183-3 2.477-3 4.50 4.183-2 8.696-3 2.856-3 1.334-3 6.157-2 1.486-2 5.179-3 2.448-3 4.75 4.573-2 9.313-3 3.178-3 1.526-3 5.832-2 1.434-2 5.142-3 2.421-3 5.00 5.048-2 1.022-2 3.628-3 1.771-3 5.320-2 1.346-2 4.936-3 2.325-3 5.25 5.649-2 1.147-2 4.210-3 2.068-3 4.787-2 1.198-2 4.461-3 2.108-3 5.50 6.436-2 1.318-2 4.941-3 2.426-3 4.018-2 9.968-3 3.738-3 1.774-3 5.75 7.481-2 1.553-2 5.888-3 2.868-3 3.167-2 7.741-3 2.904-3 1.384-3 3.75 8.886-3 2.783-3 1.046-3 6.597-4 1.716-2 6.041-3 2.074-3 1.193-3 4.00 1.299-2 3.678-3 1.504-3 9.203-4 2.233-2 6.906-3 2.468-3 1.466-3 4.25 1.910-2 5.128-3 2.143-3 1.223-3 2.826-2 7.931-3 2.983-3 1.762-3 4.50 3.006-2 7.760-3 3.173-3 1.701-3 3.477-2 9.178-3 3.585-3 2.070-3 4.75 5.178-2 1.295-2 5.180-3 2.673-3 4.128-2 1.059-2 4.236-3 2.378-3 5.00 9.534-2 2.343-2 9.287-3 4.706-3 4.635-2 1.185-2 4.796-3 2.625-3 5.25 1.778-1 4.340-2 1.713-3 8.605-3 4.800-2 1.240-2 5.048-3 2.704-3 5.50 3.167-1 7.716-2 3.040-2 1.521-2 4.503-2 1.184-2 4.832-3 2.545-3 5.75 5.217-1 1.274-1 5.023-2 2.515-2 3.808-2 1.020-2 4.169-3 2.166-3 3.75 3.884-3 1.416-3 7.906-4 1.863-3 1.030-3 8.023-4 4.00 3.809-3 1.442-3 8.285-4 2.060-3 1.125-3 8.630-4 4.25 3.783-3 1.548-3 8.888-4 2.202-3 1.247-3 9.306-4 4.50 3.942-3 1.764-3 1.009-3 2.322-3 1.354-3 9.867-4 4.75 4.390-3 2.152-3 1.230-3 2.396-3 1.407-3 9.957-4 5.00 5.266-3 2.790-3 1.585-3 2.372-3 1.389-3 9.480-4 5.25 6.598-3 3.642-3 2.047-3 2.195-3 1.283-3 8.485-4 5.50 8.159-3 4.521-3 2.522-3 1.860-3 1.087-3 7.044-4 5.75 9.556-3 5.222-3 2.900-3 1.434-3 8.368-4 5.371-4 3.75 5.864-4 4.221-4 5.082-4 5.039-4 4.00 4.917-4 3.813-4 4.547-4 4.501-4 4.25 4.101-4 3.488-4 3.748-4 3.664-4 4.50 3.467-4 3.149-4 2.945-4 2.801-4 4.75 3.011-4 2.797-4 2.268-4 2.078-4 5.00 2.697-4 2.472-4 1.724-4 1.525-4 5.25 2.436-4 2.179-4 1.279-4 1.111-4 5.50 2.135-4 1.881-4 9.081-5 7.878-5 5.75 1.781-4 1.564-4 6.114-5 5.335-5 3.75 1.938-4 3.037-4 4.00 1.551-4 2.389-4 4.25 1.158-4 1.717-4 4.50 8.143-5 1.157-4 4.75 5.484-5 7.485-5 5.00 3.640-5 4.734-5 5.25 2.473-5 2.968-5 5.50 1.742-5 1.860-5 5.75 1.244-5 1.197-5

 Transition n=2 n=3 n=4 n=5 Transition n=2 n=3 n=4 n=5 3.75 2.389 3.544-1 1.199-1 5.624-2 2.410 6.651-1 2.321-1 4.00 2.456 3.295-1 1.062-1 4.930-2 2.286 5.911-1 2.154-1 4.25 2.275 2.832-1 8.877-2 4.129-2 2.235 5.383-1 2.001-1 4.50 1.916 2.280-1 7.059-2 3.305-2 2.370 5.328-1 2.009-1 4.75 1.496 1.730-1 5.351-2 2.517-2 2.761 5.919-1 2.265-1 5.00 1.111 1.247-1 3.870-2 1.823-2 3.397 7.092-1 2.732-1 5.25 8.003-1 8.624-2 2.682-2 1.261-2 4.187 8.591-1 3.302-1 5.50 5.660-1 5.765-2 1.793-2 8.402-3 5.013 1.016 3.889-1 5.75 3.944-1 3.747-2 1.165-2 5.419-3 5.755 1.163 4.443-1 3.75 7.965-1 1.306-1 4.941-2 3.272-2 1.508+1 1.606 4.476-1 1.833-1 4.00 9.579-1 1.499-1 5.791-2 3.599-2 2.580+1 1.611 4.465-1 1.936-1 4.25 1.042 1.543-1 6.089-2 3.587-2 4.185+1 1.576 4.476-1 2.015-1 4.50 1.015 1.439-1 5.740-2 3.259-2 6.455+1 1.552 4.578-1 2.105-1 4.75 8.950-1 1.234-1 4.930-2 2.730-2 9.526+1 1.615 4.938-1 2.291-1 5.00 7.265-1 9.896-2 3.939-2 2.141-2 1.351+2 1.868 5.842-1 2.707-1 5.25 5.516-1 7.494-2 2.964-2 1.586-2 1.842+2 2.410 7.603-1 3.493-1 5.50 3.948-1 5.367-2 2.106-2 1.114-2 2.401+2 3.274 1.037 4.719-1 5.75 2.677-1 3.639-2 1.419-2 7.434-3 2.976+2 4.382 1.408 6.315-1 3.75 2.479-1 8.385-2 4.285-2 1.392 3.528-1 1.678-1 4.00 2.591-1 8.375-2 4.240-2 1.954 4.605-1 2.013-1 4.25 2.558-1 8.145-2 4.053-2 2.760 6.158-1 2.496-1 4.50 2.375-1 7.635-2 3.781-2 3.900 8.407-1 3.309-1 4.75 2.071-1 6.833-2 3.398-2 5.496 1.182 4.661-1 5.00 1.694-1 5.755-2 2.874-2 7.599 1.672 6.649-1 5.25 1.292-1 4.502-2 2.253-2 9.990 2.262 9.058-1 5.50 9.188-2 3.264-2 1.635-2 1.226+1 2.841 1.143 5.75 6.137-2 2.212-2 1.108-2 1.408+1 3.319 1.339 3.75 4.106-2 2.766-2 3.325-1 1.840-1 4.00 4.019-2 2.809-2 3.889-1 2.165-1 4.25 3.838-2 2.713-2 4.620-1 2.564-1 4.50 3.475-2 2.450-2 5.534-1 3.066-1 4.75 2.954-2 2.068-2 6.671-1 3.692-1 5.00 2.341-2 1.628-2 7.814-1 4.329-1 5.25 1.719-2 1.192-2 8.556-1 4.769-1 5.50 1.175-2 8.147-3 8.707-1 4.903-1 5.75 7.577-3 5.257-3 8.385-1 4.774-1 3.75 1.409-2 8.969-2 4.00 1.175-2 8.533-2 4.25 9.224-3 7.928-2 4.50 6.900-3 7.392-2 4.75 4.932-3 6.903-2 5.00 3.359-3 6.279-2 5.25 2.181-3 5.440-2 5.50 1.360-3 4.498-2 5.75 8.284-4 3.616-2

 Transition n=2 n=3 n=4 n=5 Transition n=2 n=3 n=4 n=5 3.75 5.290-1 1.237-1 4.939-2 6.360-1 2.118-1 8.235-2 4.00 5.736-1 1.268-1 4.801-2 5.308-1 1.753-1 7.071-2 4.25 6.711-1 1.371-1 5.053-2 4.177-1 1.368-1 5.659-2 4.50 8.581-1 1.667-1 6.149-2 3.131-1 1.017-1 4.293-2 4.75 1.149 2.224-1 8.346-2 2.252-1 7.260-2 3.113-2 5.00 1.526 2.990-1 1.137-1 1.566-2 4.992-2 2.164-2 5.25 1.940 3.836-1 1.467-1 1.061-2 3.325-2 1.449-2 5.50 2.344 4.653-1 1.782-1 7.073-2 2.161-2 9.418-3 5.75 2.705 5.373-1 2.063-1 4.654-2 1.378-2 5.987-3 3.75 1.099+1 2.806-1 8.756-2 4.363-2 1.591 4.914-1 1.520-1 5.736-2 4.00 1.929+1 3.409-1 1.136-1 5.476-2 1.728 4.684-1 1.437-1 5.789-2 4.25 3.097+1 4.246-1 1.480-1 6.919-2 1.735 4.092-1 1.273-1 5.383-2 4.50 4.628+1 5.629-1 1.945-1 8.892-2 1.579 3.304-1 1.053-1 4.629-2 4.75 6.536+1 8.352-1 2.694-1 1.208-1 1.310 2.503-1 8.195-2 3.715-2 5.00 8.828+1 1.374 4.034-1 1.769-1 1.008 1.806-1 6.048-2 2.803-2 5.25 1.146+2 2.301 6.265-1 2.685-1 7.297-1 1.250-1 4.251-2 1.996-2 5.50 1.430+2 3.638 9.482-1 3.996-1 5.015-1 8.311-2 2.850-2 1.347-2 5.75 1.719+2 5.299 1.354 5.658-1 3.292-1 5.312-2 1.829-2 8.677-3 3.75 8.375-1 2.070-1 7.112-2 2.596-1 7.885-2 4.162-2 4.00 1.167 2.355-1 8.076-2 3.050-1 8.766-2 4.405-2 4.25 1.675 2.874-1 9.700-2 3.263-1 9.224-2 4.461-2 4.50 2.438 3.808-1 1.296-1 3.166-1 9.066-2 4.341-2 4.75 3.529 5.415-1 1.893-1 2.824-1 8.324-2 4.007-2 5.00 4.941 7.779-1 2.781-1 2.332-1 7.090-2 3.430-2 5.25 6.494 1.058 3.829-1 1.782-1 5.558-2 2.693-2 5.50 7.930 1.328 4.830-1 1.265-1 4.019-2 1.947-2 5.75 9.070 1.548 5.637-1 8.433-2 2.714-2 1.312-2 3.75 1.837-1 8.834-2 5.952-2 3.836-2 4.00 2.230-1 1.039-1 6.338-2 4.042-2 4.25 2.829-1 1.295-1 6.369-2 4.009-2 4.50 3.657-1 1.681-1 5.925-2 3.707-2 4.75 4.717-1 2.204-1 5.112-2 3.201-2 5.00 5.799-1 2.762-1 4.083-2 2.568-2 5.25 6.564-1 3.192-1 3.011-2 1.903-2 5.50 6.864-1 3.409-1 2.062-2 1.310-2 5.75 6.807-1 3.443-1 1.330-2 8.484-3 3.75 4.146-2 2.367-2 4.00 4.233-2 1.929-2 4.25 4.377-2 1.507-2 4.50 4.673-2 1.140-2 4.75 4.971-2 8.329-3 5.00 5.011-2 5.804-3 5.25 4.685-2 3.842-3 5.50 4.115-2 2.429-3 5.75 3.499-2 1.486-3

 Transition n=2 n=3 n=4 n=5 Transition n=2 n=3 n=4 n=5 3.75 6.756-1 2.971-1 1.446-1 5.858 2.192 5.585-1 4.00 6.387-1 2.664-1 1.276-1 5.614 1.778 5.196-1 4.25 5.623-1 2.255-1 1.074-1 5.828 1.513 4.854-1 4.50 4.633-1 1.801-1 8.586-2 7.035 1.432 4.860-1 4.75 3.582-1 1.359-1 6.503-2 9.926 1.587 5.507-1 5.00 2.610-1 9.742-2 4.678-2 1.524+1 2.026 6.982-1 5.25 1.803-1 6.655-2 3.204-2 2.332+1 2.750 9.268-1 5.50 1.188-1 4.351-2 2.097-2 3.372+1 3.700 1.217 5.75 7.516-2 2.738-2 1.320-2 4.549+1 4.797 1.549 3.75 3.689 8.012-1 2.403-1 1.246-1 1.295+1 3.340 1.101 4.00 4.259 8.612-1 2.671-1 1.370-1 1.325+1 3.265 1.151 4.25 4.507 8.397-1 2.699-1 1.374-1 1.399+1 3.278 1.215 4.50 4.321 7.465-1 2.475-1 1.253-1 1.555+1 3.471 1.329 4.75 3.800 6.137-1 2.082-1 1.051-1 1.815+1 3.912 1.527 5.00 3.135 4.752-1 1.637-1 8.244-2 2.177+1 4.586 1.804 5.25 2.464 3.510-1 1.218-1 6.106-2 2.600+1 5.404 2.125 5.50 1.857 2.479-1 8.619-2 4.297-2 3.031+1 6.240 2.444 5.75 1.346 1.674-1 5.809-2 2.886-2 3.425+1 6.988 2.727 3.75 1.333 4.536-1 2.103-1 1.003+1 2.860 1.357 4.00 1.353 4.278-1 1.998-1 1.438+1 3.613 1.587 4.25 1.303 3.940-1 1.832-1 2.173+1 4.820 1.961 4.50 1.192 3.539-1 1.639-1 3.489+1 6.907 2.676 4.75 1.032 3.074-1 1.426-1 5.878+1 1.071+1 4.041 5.00 8.388-1 2.534-1 1.180-1 9.961+1 1.725+1 6.387 5.25 6.339-1 1.949-1 9.106-2 1.610+2 2.697+1 9.836 5.50 4.467-1 1.394-1 6.532-2 2.404+2 3.937+1 1.418+1 5.75 2.965-1 9.361-2 4.392-2 3.309+2 5.344+1 1.908+1 3.75 2.011-1 1.565-1 2.400 1.364 4.00 1.921-1 1.445-1 2.836 1.533 4.25 1.830-1 1.306-1 3.497 1.809 4.50 1.683-1 1.147-1 4.478 2.253 4.75 1.458-1 9.666-2 5.866 2.906 5.00 1.170-1 7.659-2 7.490 3.693 5.25 8.647-2 5.636-2 8.936 4.422 5.50 5.919-2 3.863-2 9.920 4.966 5.75 3.814-2 2.499-2 1.043+1 5.240 3.75 6.174-2 6.103-1 4.00 5.161-2 5.874-1 4.25 4.157-2 5.743-1 4.50 3.257-2 5.844-1 4.75 2.459-2 6.035-1 5.00 1.758-2 6.009-1 5.25 1.181-2 5.601-1 5.50 7.523-3 4.928-1 5.75 4.597-3 4.207-1

 Figure 1: Effective collision strengths vs. T/104. Full line, Sawey & Berrington (1993); dashed line, present results

Until now the best source of effective collision strengths has been Sawey & Berrington (1993) - hereafter SB - their data being based on a 29-state R-matrix calculation (Sawey et al. 1990). Using SB's tables one can obtain electron rate coefficients for 157 of the 171 transitions that exist between levels with principal quantum number . This tabulation is an improvement on the one produced by Berrington & Kingston (1987) - hereafter BK - from a 19-state R-matrix calculation for transitions between levels with . During a workshop for assessing atomic data, which was held in Saint Catherine's College Oxford in 1987, Pradhan (1987) had this to say about the reliability of their data: "For neutral helium the work by Berrington & Kingston, which included the n = 4 levels, is rated as having an uncertainty of 10% for excitation to the n = 2 states and an uncertainty of 30% for transitions to the n = 3 states''.

The tabulations of in BK and in SB extend only as far as 30000 degrees. There are, however, astrophysical situations where helium rate coefficients are needed at temperatures of the order of 120000 degrees (Bouret 1998); and in the domain of controlled thermonuclear research Summers (1999) tells us that experiments are being performed in which neutral helium is used as a diagnostic tool by being injected into plasmas where the electron temperature can be as high as degrees.

 Figure 2: CCC collision strengths vs. final energy in Ry

 Figure 3: CCC collision strengths vs. final energy in Ry

 Figure 4: CCC collision strengths vs. final energy in Ry

 Figure 5: CCC collision strengths vs. final energy in Ry

 Figure 6: CCC collision strengths vs. final energy in Ry

 Figure 7: CCC collision strengths vs. final energy in Ry

 Figure 8: CCC collision strengths vs. final energy in Ry

 Figure 9: CCC collision strengths vs. final energy in Ry

 Figure 10: CCC collision strengths vs. final energy in Ry

 Figure 11: CCC collision strengths vs. final energy in Ry

The CCC approximation is not subject to the restrictions which limit the R-matrix method with increasing collision energy. As a result we are able to obtain converged cross sections when . After being thermally averaged these provide rate coefficients for temperatures up to about 500000 degrees.

In Tables 2 - 5 we give effective collision strengths over the temperature range . Because of the uncertainty that pseudo resonances introduce into our cross sections at energies close to threshold, we limit the low temperature end of our tabulation to about 6000 degrees. Some values of in Tables 2 - 5 are given in the form ab, where b denotes the power of 10 by which a should be multiplied. In Fig. 1 we compare our results with those of SB for excitation from to the eight levels . For the three highest lying levels, namely , our results lie above or very close to those of SB. For all the other transitions in Fig. 1 our results lie below SB's. This is not true at very low temperatures for and . We find that has a narrow peak just above threshold energy and it is this pseudo resonance which causes to rise steeply as . There are two transitions, namely , for which the differences are of the order of a factor of 3. In Section 6 we give a physical explanation why SB's results and ours can sometimes differ by a large amount.

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