On the validity of using the TKR sum rule to place zero angle electron impact spectra on an absolute scale

The use of the Thomas—Kuhn—Reiche (TKR) sum rule to place electron impact spectra on an absolute scale is only rigorously correct if the observed intensities can be extrapolated to their first Born zero momentum transfer limit. This note investigates the error involved in using the TKR sum rule at zero scattering angle instead of zero momentum transfer. By considering an expansion of the generalized oscillator strength at zero angle it is shown that the first order correction to the TKR sum rule can be written as ($\text{1}\text{3}\text{k}{}^2$)|E_o|?($\text{4}\text{5}$)$\text{V}$_ee] for target systems randomly oriented in space where k² is the incident electron energy and |E_o| and $\text{V}$_ee are the magnitude of the total electronic energy and the electron—electron repulsive energy of the ground state of the target system respectively. Estimates based on Hartree—Fock calculations indicate a 0.27% error in the use of the TKR sum rule for F and 0.79% for Ar at an incident electron energy of 25 keV.