  The effect of random competing single-ion anisotropies in antiferromagnets
was studied using epitaxial Mn$_x$Ni$_{1-x}$F$_2$ antiferromagnetic thin film
alloys grown via molecular beam epitaxy. The crystal structure of this material
is tetragonal for all values of $x$, and the Mn sites have a magnetic easy axis
single-ion anisotropy while the Ni sites have an easy plane anisotropy
perpendicular to the Mn easy axis. Crystallographic and magnetization
measurements demonstrated that the thin film alloys were homogeneously mixed
and did not phase-separate into their constituent parts. Pure MnF$_2$ thin
films epitaxially grown on MgF$_2$ exhibited compressive strain along all three
crystallographic axes which resulted in piezomagnetic effects. The
piezomagnetism disappeared if the film was grown on a (MnNi)F$_2$ graded buffer
layer. A mean-field theory fit to the transition temperature as a function of
the Mn concentration $x$, which takes into account piezomagnetic effects, gave
a magnetic exchange constant between Mn and Ni ions of $J_{\text{MnNi}} = 0.305
\pm 0.003$~meV. Mean-field theory calculations also predicted the existence of
an oblique antiferromagnetic phase in the Mn$_x$Ni$_{1-x}$F$_2$ alloy which
agreed with the experimental data. A magnetic phase diagram for
Mn$_x$Ni$_{1-x}$F$_2$ thin film alloys was constructed and showed evidence for
the existence of two unique magnetic phases, in addition to the ordinary
antiferromagnetic and paramagnetic phases: an oblique antiferromagnetic phase,
and an emergent magnetic phase proposed to be either a magnetic glassy phase or
a helical phase. The phase diagram is quantitatively different from that of
Fe$_x$Ni$_{1-x}$F$_2$ because of the much larger single-ion anisotropy of
Fe$^{2+}$ compared to Mn$^{2+}$.
