  Polarised neutron diffraction measurements have been made on HoFeO$_3$ single
crystals magnetised in both the [001] and [100] directions ($Pbnm$ setting).
The polarisation dependencies of Bragg reflection intensities were measured
both with a high field of H = 9 T parallel to [001] at T = 70 K and with the
lower field H = 0.5 T parallel to [100] at T = 5, 15, 25~K. A Fourier
projection of magnetization induced parallel to [001], made using the $hk0$
reflections measured in 9~T, indicates that almost all of it is due to
alignment of Ho moments. Further analysis of the asymmetries of general
reflections in these data showed that although, at 70~K, 9~T applied parallel
to [001] hardly perturbs the antiferromagnetic order of the Fe sublattices, it
induces significant antiferromagnetic order of the Ho sublattices in the
$x\mhyphen y$ plane, with the antiferromagnetic components of moment having the
same order of magnitude as the induced ferromagnetic ones. Strong intensity
asymmetries measured in the low temperature $\Gamma_2$ structure with a lower
field, 0.5 T $\parallel$ [100] allowed the variation of the ordered components
of the Ho and Fe moments to be followed. Their absolute orientations, in the
180\degree\ domain stabilised by the field were determined relative to the
distorted perovskite structure,. This relationship fixes the sign of the
Dzyalshinski-Moriya (D-M) interaction which leads to the weak ferromagnetism.
Our results indicate that the combination of strong y-axis anisotropy of the Ho
moments and Ho-Fe exchange interactions breaks the centrosymmetry of the
structure and could lead to ferroelectric polarization.
