Magnetoelastic effects in multiferroic YMnO$_3$

We have investigated magnetoelastic effects in multiferroic YMnO$_3$ below the antiferromagnetic phase transition, $T_N \approx 70$ K, using neutron powder diffraction. The $a$ lattice parameter of the hexagonal unit cell of YMnO$_3$ decreases normally above $T_N$, but decreases anomalously below T$_N$, whereas the $c$ lattice parameter increases with decreasing temperature and then increases anomalously below T$_N$. The unit cell volume also undergoes an anomalous contraction below $T_N$. By fitting the background thermal expansion for a non-magnetic lattice with the Einstein-Gr\"uneisen equation, we determined the lattice strains $\Delta a$, $\Delta c$ and $\Delta V$ due to the magnetoelastic effects as a function of temperature. We have also determined the temperature variation of the ordered magnetic moment of the Mn ion by fitting the measured Bragg intensities of the nuclear and magnetic reflections with the known crystal and magnetic structure models and have established that the lattice strain due to the magnetoelastic effect in YMnO$_3$ couples with the square of the ordered magnetic moment or the square of the order parameter of the antiferromagnetic phase transition.