  We propose an experiment for measuring the effective Casimir pressure between
two parallel SiC plates with aligned nuclear spins. The prospective constraints
on an axion-neutron coupling constant for both hadronic and GUT axions are
calculated using the process of one-axion exchange. For this purpose, a general
expression for the additional pressure arising between two polarized plates due
to the exchange of one axion between their constituent fermions is derived. We
demonstrate that only the polarization component perpendicular to the plates
contribute to the pressure. The obtained pressure can be both repulsive and
attractive depending on whether the polarizations of both plates are
unidirectional or directed in opposite directions. It is shown that although
the constraints on an axion-electron coupling obtained in the case of
magnetized plates are not competitive, the constraints on an axion-neutron
coupling found for plates with polarized nuclear spins are of the same order of
magnitude of those obtained previously for the GUT axions alone using the
process of two-axion exchange. The proposed experiment allows also to
strengthen the presently known constraints on the axion-neutron coupling
constants of GUT axions by using both processes of one- and two-axion exchange.
