  The asymptotic staying probability P in the Landau-Zener effect with
interaction is analytically investigated at fast sweep, epsilon = pi Delta^2/(2
hbar v) << 1. We have rigorously calculated the value of I_0 in the expansion P
=~ 1 - epsilon + epsilon^2/2 + epsilon^2 I_0 for arbitrary couplings and
relative resonance shifts of individual tunneling particles. The results
essentially differ from those of the mean-field approximation. It is shown that
strong long-range interactions such as dipole-dipole interaction (DDI) generate
huge values of I_0 because flip of one particle strongly influences many
others. However, in the presence of strong static disorder making resonance for
individual particles shifted with respect to each other the influence of
interactions is strongly reduced. In molecular magnets the main source of
static disorder is the coupling to nuclear spins. Our calculations using the
actual shape of the Fe-8 crystal studied in the the Landau-Zener experiments
[Wernsdorfer et al, Europhys. Lett. 50, 552 (2000)] yield I_0 that is in a good
agreement with the value extracted from the experimental data.
