  The double coherent and non-coherent diffractive production of heavy quark -
antiquark pairs ($Q \bar{Q}$) in heavy ion scattering at high energies (LHC) is
considered. The total and differential cross sections of these processes with
the formation of $c \bar{c}$ and $b \bar{b}$ pairs in $pp$, $CaCa$ and $PbPb$
collisions are evaluated. The contribution of the considered mechanisms is a
few per cent of the number of heavy quark - antiquark pairs obtained in the
processes of hard (QCD) scattering, and it will be taken into account in the
registration of $c$, $b$ quarks or, for instance, in the study of the heavy
quarkonia suppression effects in Quark - Gluon Plasma, in the search for
intermediate mass Higgs bosons and so on. It is shown that the cross section of
the coherent scattering process is great enough. This makes it suitable for
studying collective effects in nuclear interactions at high energies. An
example of such effects is given: large values of the invariant mass of a $Q
\bar{Q}$pair, $M_{Q \bar{Q}} \gsim 100$ GeV, in association with a large
rapidity gap between diffractive jets $\Delta \eta > 5$.
