Electron-Impact Cross Sections for Dipole- and Spin-Allowed Excitations of Hydrogen, Helium, and Lithium

Electron-impact excitation cross sections are presented for the dipole- and spin allowed transitions from the ground states to the np 2P states for hydrogen and lithium, and to the 1snp 1P states for helium, n = 2 through 10. Two scaling formulas developed earlier by Kim [Phys. Rev. A 64, 032713 (2001)] for plane-wave Born cross sections are used. The scaled Born cross sections are in excellent agreement with available theoretical and experimental data.


Introduction
We have scaled Plane-Wave Born (PWB) cross sections to calculate dipole-and spin-allowed excitation cross sections from the ground state of neutral hydrogen, helium, and lithium. The scaling method was developed by one of us [1], and uses two simple scaling formulas to convert PWB excitation cross sections into reliable cross sections comparable to the most accurate theoretical or experimental data available for dipole-allowed transitions. The PWB cross sections are calculated from uncorrelated wave functions, and the scaling requires only the binding energy B of the electron being excited, the excitation energy E , and an accurate dipole oscillator strength f for the transition. The oscillator strength is needed only if electron correlation strongly affects the f value, i.e., when the wave functions used to calculate the PWB cross section are not accurate. Simplicity of the method to scale PWB cross sections allows us to generate a large number of cross sections reliably and quickly.
In this paper, we present calculated excitation cross sections for hydrogen from the 1s 2 S ground state to the np 2 P excited states. For helium, the cross sections are given for excitations from the 1s 2 1 S ground state to the 1snp 1 P excited states. For lithium, the results are given for excitations from the 1s 2 2s 2 S ground state to the 1s 2 np 2 P states. In all cases, the values of n are from n = 2 through 10.

Outline of Theory
A PWB cross section for electron-impact excitation, PWB , has the form where T is the incident electron energy, a 0 is the Bohr radius (0.529 Å), and R is the Rydberg energy (13.61 eV). The F PWB (T ) is the collision strength (different from the standard definition by a multiplicative constant).
The first scaling method, BE scaling, replaces T in the denominator of Eq. (1) by T + B + E , i.e., This scaling is similar to a scaling for ionization cross sections used earlier by Burgess [2], who shifted the incident energy T by B+U , where U is the kinetic energy of the target electron. However, in the BE scaling adopted by Kim [1] for excitation cross sections, T is shifted by B+E . The BE scaling not only changes the magnitude but also the shape of the original PWB cross sections. The BE scaling corrects the deficiency in the collision theory; i.e., the use of the PWB approximation.
The second scaling formula, the f scaling, multiplies the entire cross section by the ratio of an accurate f value to the less accurate f value calculated by the actual wave functions used to generate the unscaled PWB cross sections: ( where f sc is the single configuration (or uncorrelated) f value and f accu is the more accurate value obtained from correlated (or multiconfiguration) wave functions or from a reliable experiment. Accurate f values are frequently available [3]. The f scaling compensates for the inadequacy of the wave functions when electron correlation effect is significant. The BE and f scalings may be applied consecutively, i.e., where BEsc is the BE-scaled PWB cross section calculated from single-configuration wave functions. Kim has shown many examples [1] in which the BE scaling alone or in combination with the f scaling transformed PWB cross sections for dipole-allowed and spinallowed excitations into reliable cross sections comparable to the convergent close coupling (CCC) method [4] or accurate experiments.
In reality, electron-impact excitation cross sections of atoms have resonances in the vicinity of the excitation thresholds caused by the formation of transient compound states between the incident electron and the target atom. First-order perturbation theories such as the PWB approximation cannot account for such compound states, and hence the present scaled cross sections do not exhibit any resonances.
The numerical data in Tables 1, 2, and 3 can easily be extended to higher incident energies by using the well known Bethe formula [5] for the plane-wave Born approximation for fast (but nonrelativistic) incident electrons. In our notation, the asymptotic expression becomes: [a ln(T /R )+b+cR /T ](f accu /f sc ), (5) where a, b, and c are dimensionless constants. Equation (5) should be used for T > 3 keV. The values of a , b , and c are included in Tables 1, 2, and 3. Note that a relativistic form (5) of Eq. (5) should be used for T > 10 keV.

Theoretical Results
We present the calculated cross sections for hydrogen, helium, and lithium in Tables 1-3. Our PWB cross sections were generated from single configuration Dirac-Fock wave functions. The calculated cross sections are compared to other theories and experiments in Figs. 1-7.
The CCC results for these elements are from the web site of Bray [6]. The experimental results by Sweeney et al. [7] for hydrogen include all dipole-allowed and dipole-forbidden states of hydrogen for each n , and hence are higher than the cross sections for just the dipole-allowed excitations.
The ionization energies B and the excitation energies E are all experimental values. Only BE scaling is needed for hydrogen as its exact wave functions are known. The accurate f values for helium have been obtained from the detailed variational calculations of Drake [8]. The f value for the 2s -2p transition in lithium is from the calculations of Yan et al. [9], while the values for the 2s -np transitions, n = 3 through 7, are from the non-relativistic multiconfiguration calculations including core polarization by Qu et al. [10]. For the 2s -8p , 9p , and 10p excitations of lithium, we extrapolated f (n *) 3 of Qu et al. [10] from n = 5 through n = 7, where n * is the experimental effective principal quantum number of quantum defect theory. We had found that the f values by Qu et al. for the 8p and 9p transitions were inconsistent with their values for n < 8. The extrapolation of f (n *) 3 is shown in Fig. 8 through n * ≈ 17 and is given by the expression: f (n *) 3 = 0.343 + 0.0283/(n *) + 0.533/(n *) 2 -6.289/(n *) 3 .
Beyond n * ≈ 17, the formula begins to break down but the actual curve should remain flat. At the ionization limit (n * → ϱ), the value of f (n *) 3 is extrapolated to be 0.345.
It is apparent that for all cases where experimental data and CCC results are available, the scaled PWB cross sections give values that are in good agreement.   [7]. The other results are as in Fig. 1.   Fig. 3. Hydrogen 1s -4p electron-impact excitation cross sections. The symbols are as in Fig. 2.  Fig. 4. Helium 1s 2 1 S-1s 2p 1 P electron-impact excitation cross sections. The solid curve is the BEf -scaled plane-wave Born cross section, the open circles are recent CCC results of Merabet et al. [12], and the filled circles are previous CCC results from the web site of Bray and Ralchenko [6]. Experimental results are the recommended values of Trajmar et al. [16] (open squares), Westerveld et al. [14] (triangles), Shemansky et al. [15] (pluses), recommended values of de Heer et al. [13] (crosses), and results of Merabet et al. [12] (diamonds).