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Calculations of the cross sections of inelastic electron scattering $(e,e')$on a nucleus in the multi-keV energy range strongly depend on the descriptionof the screening of the nuclear Coulomb potential as well as on the deformationof the wave functions of the incoming and outgoing electron in the vicinity ofthe 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 areal challenge to measure such low cross sections. In this study, wedemonstrate that inelastic electron scattering is the main nuclear excitationmechanism in a $^{181}$Ta target irradiated with a new intense 10 - 30 keVelectron beam produced with a biased laser-plasma. Calculations show thatthrough the detection of conversion electrons, it should be possible to measurethe nuclear excitation yields. The effect of electron beam heating and ofplasma deposition on the tantalum target are quantified, thus allowing thedimensioning of a possible experimental configuration to study $(e,e')$processes in this range of energy for the first time.

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