  Neutrino-nucleus scattering $\nu A\to \nu A$, in which the nucleus conserves
its integrity, is considered. We show that elastic interactions keeping the
nucleus in the same quantum state lead to a quadratic enhancement of the
corresponding cross-section in terms of the number of nucleons. Meanwhile, the
cross-section of inelastic processes in which the quantum state of the nucleus
is changed, essentially has a linear dependence on the number of nucleons.
These two classes of processes are referred to as coherent and incoherent,
respectively. The coherent and incoherent cross-sections are driven by factors
$|F_{p/n}|^2$ and $(1-|F_{p/n}|^2)$, where $|F_{p/n}|^2$ is a proton/neutron
form-factor of the nucleus, averaged over its initial states. The coherent
cross-section formula used in the literature is revised and corrections
depending on kinematics are estimated. As an illustration of the importance of
the incoherent channel we considered three experimental setups with different
nuclei. Experiments attempting to measure coherent neutrino scattering by
solely detecting the recoiling nucleus, as is typical, might be including an
incoherent background that is indistinguishable from the signal if the
excitation gamma eludes its detection. However, as is shown here, the
incoherent component can be measured directly by searching for photons released
by the excited nuclei inherent to the incoherent channel. For a beam experiment
these gammas should be correlated in time with the beam, and their higher
energies make the corresponding signal easily detectable at a rate governed by
the ratio of incoherent to coherent cross-sections. The detection of signals
due to the nuclear recoil and excitation gammas provides a more sensitive
instrument in studies of nuclear structure and possible signs of new physics.
