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Neutrino-nucleus $\nu A\to \nu A$ and antineutrino-nucleus $\bar\nu A\to\bar\nu A$ interactions, when the nucleus conserves its integrity, arediscussed with coherent (elastic) and incoherent (inelastic) scattering regimestaken into account. In the first regime the nucleus remains in the same quantumstate after the scattering and the cross-section depends on the quadraticnumber of nucleons. In the second regime the nucleus changes its quantum stateand the cross-section has an essentially linear dependence on the number ofnucleons. The coherent and incoherent cross-sections are driven by a nuclearnucleon form-factor squared $|F|^2$ term and a $(1-|F|^2)$ term, respectively.One has a smooth transition between the regimes of coherent and incoherent(anti)neutrino-nucleus scattering. Due to the neutral current nature theseelastic and inelastic processes are indistinguishable if the nucleus recoilenergy is only observed. One way to separate the coherent signal from theincoherent one is to register $\gamma$ quanta from deexcitation of the nucleusexcited during the incoherent scattering. Another way is to use a verylow-energy threshold detector and collect data at very low recoil energies,where the incoherent scattering is vanishingly small. In particular, for${}^{133}\text{Cs}$ and neutrino energies of 30--50 MeV the incoherentcross-section is about 15-20\% of the coherent one. Therefore, the COHERENTexperiment (with ${}^{133}\text{Cs}$) has measured the coherent elasticneutrino nucleus scattering (CE$\nu$NS) with the inelastic admixture at a levelof 15-20\%, if the excitation $\gamma$ quantum escapes its detection.

On coherent neutrino and antineutrino scattering off nuclei