  Ferroelectricity has been found to occur in several insulating systems, such
as TbMnO$_3$ (TMO) and Ni$_3$V$_2$O$_8$ (NVO) which have more than one phase
with incommensurately modulated long-range magnetic order. Here we give a
phenomenological model which relates the symmetries of the magnetic structure
as obtained from neutron diffraction to the development and orientation of a
spontaneous ferroelectric moment induced by the magnetic ordering. This model
leads directly to the formulation of a microscopic spin-phonon interaction
which gives explains the observed phenomena. The results are given in terms of
gradients of the exchange tensor with respect to generalized displacements for
the specific example of NVO. It is assumed that these gradients will now be the
target of first-principles calculations using the LDA or related schemes.
