  Using high resolution powder x-ray and neutron diffraction experiments, we
determined the off-centered displacement of the ions within a unit cell and
magnetoelectric coupling in nanoscale BiFeO$_3$ ($\approx$20-200 nm). We found
that both the off-centered displacement of the ions and magnetoelectric
coupling exhibit nonmonotonic variation with particle size. They increase as
the particle size reduces from bulk and reach maximum around 30 nm. With
further decrease in particle size, they decrease precipitously. The
magnetoelectric coupling is determined by the anomaly in off-centering of ions
around the magnetic transition temperature ($T_N$). The ions, in fact, exhibit
large anomalous displacement around the $T_N$ which is analyzed using group
theoretical approach. It underlies the nonmonotonic particle-size-dependence of
off-centre displacement of ions and magnetoelectric coupling. The nonmonotonic
variation of magnetoelectric coupling with particle size is further verified by
direct electrical measurement of remanent ferroelectric hysteresis loops at
room temperature under zero and $\sim$20 kOe magnetic field. Competition
between enhanced lattice strain and compressive pressure appears to be causing
the nonmonotonic particle-size-dependence of off-centre displacement while
coupling between piezo and magnetostriction leads to nonmonotonicity in the
variation of magnetoelectric coupling.
