Second Born Zero Angle Differential Cross Sections in the High Energy Limit for the Electron Impact Excitation of He: The 23S ← 11S and 23P ← 11S Transitions

The high energy limit of the inelastic differential cross section for the excitation of the 2³S and 2³P states of He from the ground state in the forward direction is investigated by expanding the T matrix in powers of V about zero scattering angle and considering all terms proportional to V and V² where V is the interaction potential. It is shown that the leading terms in a power series in the reciprocal of the incident energy in this second Born infinite channel close coupling theory can qualitatively account for the observed enhancement of the exchange cross section in the forward direction over the predictions of the first Born theory. The ratio of the square of the leading second Born term to the square of the leading first Born term is shown to be proportional to the incident energy for the $2^3{\text{S ← 1}}^{1}S$ transitions and of the same order in the incident energy as the first Born form in the case of the $2^3{\text{P ← 1}}^{1}S$ transition. The value of the ratio at 500 eV for the $2^3{\text{S ← 1}}^{1}P$ transition is estimated to be close to 100 and is relatively independent of the choice of an effective energy loss parameter introduced to carry out the sum over intermediate states.