  The paper presents a novel approach to the description of the nonrelativistic
weak interaction of a massive neutral particle (lepton) and a nucleus, in which
the latter retains its integrity. The cross section of such a process is a sum
of the elastic (or coherent) contribution, when the nucleus remains in its
original state, and the inelastic (incoherent) contribution, when the nucleus
is in an excited state. Smooth transition from elastic scattering to inelastic
scattering is governed by the dependence of the nuclear form factors on the
momentum transferred to the nucleus. The intensity of the weak interaction is
set by the parameters that determine the contributions to the probability
amplitude from the scalar products of the leptonic and nucleon currents. The
resulting expressions are of interest, at least in the problem of direct
detection of neutral massive weakly interacting particles of dark matter, since
in this case, in contrast to the generally accepted approach, both elastic and
inelastic processes are simultaneously considered. It is shown that the
presence of the inelastic contribution accompanied by emission of
characteristic radiation (photons) from the deexcitation of the nucleus turns
out to be decisive when the coherent cross section is strongly suppressed or
cannot be detected. Therefore in order to extract maximum information about
dark matter particles, one should plan experiments aimed at the direct
detection of dark matter particles in a setting that allows one to detect both
the recoil energy of the nucleus and the gamma quanta from the deexcitation of
the nucleus.
