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Expected yields of $^{181}$Ta (e,e') $^{181}$Ta$^*$ in the multi-keV range with a plasma-cathode electron beam

Calculations of the cross sections of inelastic electron scattering $(e,e')$ on a nucleus in the multi-keV energy range strongly depend on the description of the screening of the nuclear Coulomb potential as well as on the deformation of the wave functions of the incoming and outgoing electron in the vicinity of the nucleus. These cross sections are evaluated at values lower than $10^{-30}$ cm$^2$ which vary by several orders of magnitude according to the models. Experimental measurements would be required to constrain the models but it is a real challenge to measure such low cross sections. In this study, we demonstrate that inelastic electron scattering is the main nuclear excitation mechanism in a $^{181}$Ta target irradiated with a new intense 10 - 30 keV electron beam produced with a biased laser-plasma. Calculations show that through the detection of conversion electrons, it should be possible to measure the nuclear excitation yields. The effect of electron beam heating and of plasma deposition on the tantalum target are quantified, thus allowing the dimensioning of a possible experimental configuration to study $(e,e')$ processes in this range of energy for the first time.