1 Introduction

Helium has been of interest to astronomers for over 130 years ever since it was discovered spectroscopically in the Sun's atmosphere. The bright yellow $\rm D^3$ chromospheric line, which Lockyer (1869a,b,c) saw in full sunlight near the limb on the 20th of October 1868, was later identified with the $\rm 4\,^3D \to 3\,^3P$ transition (587.6 nm) in He I.

Bartschat (1998) gives a concise summary of the recent evolution of our understanding of electron excitation of helium, the process to which the present paper is devoted; the list of references he provides is particularly useful.

Quantum mechanics provides the means of investigating how atomic helium behaves when bombarded by electrons, and there has been much effort made to quantify this since the advent of electronic computers. The convergent close-coupling (CCC) method for calculating electron-helium collisions has been developed by Fursa & Bray (1995), and has proved to be one of the most successful to date. Astrophysicists are interested in obtaining rate coefficients for collisional excitation and it is with these quantities that the present paper is concerned.

Bartschat (1998, 1999) has treated some of the transitions dealt with here using the R-matrix with pseudo states (RMPS) approximation (Bartschat et al. 1996). Since his results are for electron energies not exceeding 40 eV with respect to the ground state, they can only provide reliable rate coefficients for temperatures below 20 thousand degrees. Bartschat's cross sections have, however, one advantage over ours in that they are delineated in great detail by the use he made of a very fine energy mesh.

The temperature dependent rate coefficient $q(i \to j)$ for a transition between atomic terms with indices i and j and energy separation E_ij is given by

$$% q(i \to j) = {2 \pi^{1/2} a_0 \hbar \over m_{\rm e}} \Bigl... ...1/2} {\rm exp}(-E_{ij}/kT) {{\sl\Upsilon}(i-j)\over \omega_i}$$ (1)

where $2 \pi^{1/2} a_0\, \hbar\, m_{\rm e}^{-1} = 3.610 \,\, 10^{-24}\,\,{\rm m^3\,\,s^{-1}}$. For energies we use the Rydberg unit, which has the value ${\rm Ry} = 13.6058$ eV, while the Boltzmann constant is given by $k = 8.617 \ 10^{-5}\,\,~{\rm eV\,\,deg^{-1}}$, the temperature T being in degrees Kelvin. The factor $\omega_i = (2S_i+1) (2L_i+1)$ is the statistical weight of the i-th term or level. The effective collision strength ${\sl\Upsilon}(i-j)$ in Eq. (1) was first introduced by Seaton (1953) who defined it as follows:

$$% {\sl\Upsilon}(i-j) = \int_0^{\infty} {\sl\Omega}(i-j) \, {\rm exp}(-E_j/kT)\, {\rm d}(E_j/kT)$$ (2)

where E_j is the energy of the colliding electron after excitation has occurred. The energy dependent collision strength ${\sl\Omega}(i-j)$ and cross section $Q(i \to j)$ are related as follows:

$$% Q(i \to j) = {{\pi\, {\sl\Omega}(i-j)} \over {\omega_i\, k_i^2}}$$ (3)

where k_i is the wave number of the colliding electron incident on the target atom in level i.